**1. Introduction**

The process of the coating deposition is very complex and undoubtedly stochastic [1–7]. Accordingly, the modeling of the coating deposition process is also a complex and many-sided challenge, and in some cases, a probabilistic approach is required to meet this challenge [3–7]. Initially, one of the key tasks for the modeling of the processes under consideration was to predict a thickness of a coating, including a coating deposited on a complex-shape surface. To meet the above challenges, physical and geometric methods were applied, which made it possible to achieve the certain prediction accuracy. In particular, the studies proposed the coating thickness uniformity model [8,9], the cosine analytical and generic computational model [10], and the inverse power law relationship between thickness and source-to-substrate distance was considered [11]. Recent studies also applied true line-of-sight models, based on the inverse square law and also taking into account the effect of shadow masks [12]. In [13], the computational level set method was also applied for tracking topographic change of a surface.

The traditional approaches to the modeling of mass transfer processes, in particular, the coating deposition processes, have significant limitations when described in micro- and mesosystems [14,15]. In connection with the above, the processes are often modeled on micro- nano- and atomic levels, due to the integration of systems with a large number of particles, the motion of which is described, in particular, by equations of molecular dynamics of Newtonian mechanics. A significant problem of these methods is the existing limitation of the maximum number of particles in a system due to the limit of computing power.

Approaches of molecular dynamics are used to model the deposition process [16]. By simulating the interaction of individual atoms, this method makes it possible to predict the morphology, thermal and electronic properties, and also internal stresses of the coating being deposited. When the above method is applied, the forces of interatomic interaction are presented in the form of classical potential forces. The configuration sets obtained in the course of the calculations using the method of molecular dynamics are distributed in accordance with a certain statistical distribution function, for example, a one corresponding to microcanonical distribution. The exact information concerning the trajectories of system particles at large time intervals is not necessary to obtain results of a macroscopic (thermodynamic) nature [17]. The principals of molecular dynamics can be applied to simulate the coating deposition process using dynamic equations and pressure and temperature boundary conditions for substrate and coating surfaces, taking into account the results of the preliminary study of stresses in the coating.

In particular, to model the coating deposition process, the Particle-in-Cell Monte Carlo simulation method [3,4], with both modeling options of simpler 2D [5] and more complex 3D [6–8], requiring advanced computing power, is applied. During the comparison of 2D and 3D models, it was found that the main difference between two-dimensional and three-dimensional options of modeling was consideration of the effect of propagating plasma waves that had a significant influence on the properties of particle transport in discharges limited by a magnetic field. Two-dimensional modeling provides useful qualitative information about the general trends and mechanisms of magnetron discharges, while three-dimensional modeling makes it possible to quantitatively describe the electron transfer and current–voltage characteristics [18]. Based on the results of 2D modeling using the Monte Carlo algorithm, the formation of a coating with feather-like hierarchical structure was demonstrated in [19]. The results of the study focused on the development of a columnar coating structure have proved that the average distance between columns first increases sharply and then stabilizes due to cessation of the growth and regeneration of the structure. The Monte Carlo methods were applied for 2D modeling of the process of micropore formation in a coating, given the stochastic nature of the process [20]. Approaches of computational fluid dynamics were also applied, they give a much rougher description of the process, but at the same time contribute to a noticeable reduction in the amount of calculations, thus making it possible to introduce modeling in the 3D geometry with high-scale detailing [21,22]. A more detailed analysis of various methods for modeling the coating deposition process can be found in [23]. Thus, the most accurate and complete simulation of the deposition process should take into account the stochastic nature of this process and consider the molecular level, because a coating is formed precisely at this level. Several studies [24–26] consider a cluster, hierarchical structure of the coating, deposited through various methods, and the influence of the deposition process parameters on the formation of such structure. However, the influence of the parameters of the nanolayer coating structure on the formation of hierarchical structures in the coating requires an additional study.
